This is the proof given in Kreszig book.
3.1-6 Hilbert sequence space $l^2$. The space $l^2$ (cf. 2.2-3) is a Hilbert space with inner product defined by $$ \langle x, y\rangle=\sum_{j=1}^\infty \xi_j \bar{\eta}_j . $$ Convergence of this series follows from the Cauchy-Schwarz inequality (11), Sec. $1.2$, and the fact that $x, y \in l^{2}$, by assumption. We see that (8) generalizes (6). The norm is defined by $$ \|x\|=\langle x, x\rangle^{1 / 2}=\left(\sum_{j=1}^{\infty}\left|\xi_{j}\right|^{2}\right)^{1 / 2} . $$ Completeness was shown in $1.5-4$.
I have problem with the proof that the series converges from Cauchy Schwarz inequality. Cauchy Schwarz inequality talks about the absolute convergence of the series. Is it because every absolutely convergent series is convergent in complex plane?